ParameterEstimation-4

# ParameterEstimation-4 - What Are Parameters Consider some...

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What Are Parameters? Consider some probability distributions: Ber(p) Poi( l ) Multinomial(p 1 , p 2 , . .., p m ) Uni(a, b) Normal( m , 2 ) Etc. Call these “parametric models” Given model, parameters yield actual distribution Usually refer to parameters of distribution as Note that that can be a vector of parameters = p = l = (p 1 , p 2 , . .., p m ) = (a, b) = ( m , 2 ) Why Do We Care? In real world, don’t know “true” parameters But, we do get to observe data o E.g., number of times coin comes up heads, lifetimes of disk drives produced, number of visitors to web site per day, etc. Need to estimate model parameters from data “Estimator” is random variable estimating parameter Want “point estimate” of parameter Single value for parameter as opposed to distribution Estimate of parameters allows: Better understanding of process producing data Future predictions based on model Simulation of processes Recall Sample Mean Consider n I.I.D. random variables X 1 , X 2 , . .. X n X i have distribution F with E[X i ] = m and Var(X i ) = 2 We call sequence of X i a sample from distribution F Recall sample mean: where Recall variance of sample mean: Clearly, sample mean X is a random variable n i i n X X 1 ] [ X E n X 2 ) ( Var Sampling Distribution Note that sample mean X is random variable “Sampling distribution of mean” is the distribution of the random variable X Central Limit Theorem tells us sampling distribution of X is approximately normal when sample size, n o Rule of thumb for “large” n : n > 30, but larger is better (> 100) o Can use CLT to make inference about sample mean

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ParameterEstimation-4 - What Are Parameters Consider some...

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